00000151.htm

134 VII SPACES OF CONTINUOUS FUNCTIONS

uniform convergence of the sequence in A. Similarly, we say that a series
(u_n) which converges in ^_F(A) to a sum s is uniformly convergent in A to the
sum s. If F is a Banach space, it follows from (7.1.3) that in order that a
series (u_m) in ^(A) be uniformly convergent, a necessary and sufficient
condition is that, for any e > 0, there exist an integer n₀ such that, for
i and any t e A, we have

From (7.13) and (5.3.2) it follows that if F is a Banach space and if a series
(z/J of bounded functions is such that the series (\\u_n\\) converges in R, then
the series (u_n) is uniformly convergent; moreover, for each t e A, since
\\u_n(t)\\ < ||wj the series (u_a(t)) is absolutely convergent in F. However,
these two properties do not imply that the series (\\u_tt\\) is convergent; to
avoid misunderstandings, we therefore say that the series («_n) is normally
convergent in #_F(A) if the series (||wj) converges. We define similarly a
normally summable family (w_A)_AeL in J*_F(A) (L denumerable, cf. Section 5.3).

PROBLEMS

1. In the space ^_R(R), let u_n be the function equal to l/n for n ^ t < n 4-1, to 0 for other
values of t. Show that the series (u_n) is uniformly and commutatively convergent
(Section 53, Problem 4) and that for every / e R, the series (u_n(t)) is absolutely con-
vergent, but that (u_n) is not normally convergent.

2. Let A be any set; show that the mapping w-^-sup u(t) of ^n(A) into R is continuous.

t € A

3. Let E be a metric space, F a normed space; show that the set of all mappings /e ^_F(E)
whose oscillation (Section 3.14) at every point of E is at most equal to a given number
a > 0, is closed in the space ^_F(E). j

2. SPACES OF BOUNDED CONTINUOUS FUNCTIONS

Let E now be a metric space; we denote by ^V(E) the vector space of
all continuous mappings of E into the normed space F, by ^(E) the set
of all bounded continuous mappings of E into F. We note that if E is compact,
*?(E) = «p(E) by (3.17.10). In general we have «f(E) = ^_F(E) n J_F(E).
We will consider #£(E) as a normed subspace of ^_F(E), unless the contrary
is explicitly stated. If F is finite dimensional, in the decomposition (7.1.2.1)
/is continuous if and only if each of the /^ is continuous (see (3.20.4) and
(5.4.2)). The remarks preceding (7.1.2) then show that in such a case, «£(E)
is a topological direct sum of a finite number of subspaces, each of which



	134 VII SPACES OF CONTINUOUS FUNCTIONS uniform convergence of the sequence in A. Similarly, we say that a series (u_n) which converges in ^_F(A) to a sum s is uniformly convergent in A to the sum s. If F is a Banach space, it follows from (7.1.3) that in order that a series (u_m) in ^(A) be uniformly convergent, a necessary and sufficient condition is that, for any e > 0, there exist an integer n₀ such that, for i and any t e A, we have

	From (7.13) and (5.3.2) it follows that if F is a Banach space and if a series (z/J of bounded functions is such that the series (\\u_n\\) converges in R, then the series (u_n) is uniformly convergent; moreover, for each t e A, since \\u_n(t)\\ < \|\|wj the series (u_a(t)) is absolutely convergent in F. However, these two properties do not imply that the series (\\u_tt\\) is convergent; to avoid misunderstandings, we therefore say that the series («_n) is normally convergent in #_F(A) if the series (\|\|wj) converges. We define similarly a normally summable family (w_A)_AeL in J*_F(A) (L denumerable, cf. Section 5.3).

	PROBLEMS

	1. In the space ^_R(R), let u_n be the function equal to l/n for n ^ t < n 4-1, to 0 for other values of t. Show that the series (u_n) is uniformly and commutatively convergent (Section 53, Problem 4) and that for every / e R, the series (u_n(t)) is absolutely con- vergent, but that (u_n) is not normally convergent. 2. Let A be any set; show that the mapping w-^-sup u(t) of ^n(A) into R is continuous. t € A 3. Let E be a metric space, F a normed space; show that the set of all mappings /e ^_F(E) whose oscillation (Section 3.14) at every point of E is at most equal to a given number a > 0, is closed in the space ^_F(E). j

	2. SPACES OF BOUNDED CONTINUOUS FUNCTIONS

	Let E now be a metric space; we denote by ^V(E) the vector space of all continuous mappings of E into the normed space F, by ^(E) the set of all bounded continuous mappings of E into F. We note that if E is compact, ?(E) = «p(E) by (3.17.10). In general we have «f(E) = ^_F(E) n J_F(E). We will consider #£(E) as a normed* subspace of ^_F(E), unless the contrary is explicitly stated. If F is finite dimensional, in the decomposition (7.1.2.1) /is continuous if and only if each of the /^ is continuous (see (3.20.4) and (5.4.2)). The remarks preceding (7.1.2) then show that in such a case, «£(E) is a topological direct sum of a finite number of subspaces, each of which